Oriented projective geometry explained

Oriented projective geometry is an oriented version of real projective geometry.

Whereas the real projective plane describes the set of all unoriented lines through the origin in R3, the oriented projective plane describes lines with a given orientation. There are applications in computer graphics and computer vision where it is necessary to distinguish between rays light being emitted or absorbed by a point.

Elements in an oriented projective space are defined using signed homogeneous coordinates. Let

n
R
*
be the set of elements of

Rn

excluding the origin.
  1. Oriented projective line,

T1

:

(x,w)\in

2
R
*
, with the equivalence relation

(x,w)\sim(ax,aw)

for all

a>0

.
  1. Oriented projective plane,

T2

:

(x,y,w)\in

3
R
*
, with

(x,y,w)\sim(ax,ay,aw)

for all

a>0

.

These spaces can be viewed as extensions of euclidean space.

T1

can be viewed as the union of two copies of

R

, the sets (x,1) and (x,-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise

T2

can be viewed as two copies of

R2

, (x,y,1) and (x,y,-1), plus one copy of

T

(x,y,0).

An alternative way to view the spaces is as points on the circle or sphere, given by the points (x,y,w) with

x2+y2+w2=1.

Oriented real projective space

Let n be a nonnegative integer. The (analytical model of, or canonical) oriented (real) projective space or (canonical) two-sided projective space

Tn

is defined as

Tn=\{\{λZ:λ\inR>0\}:Z\inRn+1\setminus\{0\}\}=\{R>0Z:Z\inRn+1\setminus\{0\}\}.

Here, we use

T

to stand for two-sided.

Distance in oriented real projective space

Distances between two points

p=(px,py,pw)

and

q=(qx,qy,qw)

in

T2

can be defined as elements

((pxqw-qx

2+(p
p
y

qw-qy

2,sign(p
p
w

qw)(pw

2)
q
w)
in

T1

.

Oriented complex projective geometry

See also: Complex projective space. Let n be a nonnegative integer. The oriented complex projective space

{CP

}^n_ is defined as

{CP

}^n_=\=\. Here, we write

S1

to stand for the 1-sphere.

See also

References